10
156
(K10n
32
)
A knot diagram
1
Linearized knot diagam
7 5 8 10 8 1 4 5 3 2
Solving Sequence
3,8 4,5
6 9 10 2 7 1
c
3
c
5
c
8
c
9
c
2
c
7
c
1
c
4
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4u
9
− 2u
8
− u
7
+ 9u
6
− 29u
5
− u
4
+ 13u
3
+ 51u
2
+ 27b − 19u + 8,
− 11u
9
+ 8u
8
+ 4u
7
− 9u
6
− 46u
5
+ 31u
4
+ 29u
3
+ 12u
2
+ 27a − 32u − 5, u
10
+ u
7
+ 5u
6
− u
3
+ u
2
− u + 1i
I
u
2
= hu
2
+ b + 1, u
3
+ a + 2u + 1, u
5
+ 2u
3
+ u
2
+ 1i
I
u
3
= h−3646u
11
+ 4692u
10
+ ··· + 3395b + 12871, 24747u
11
− 25539u
10
+ ··· + 16975a − 130862,
u
12
− u
11
+ 2u
10
− 3u
9
+ 6u
8
− 3u
7
+ 9u
6
− 5u
5
− 2u
4
− 8u
2
− 6u − 1i
* 3 irreducible components of dim
C
= 0, with total 27 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= h−4u
9
− 2u
8
+ · · · + 27b + 8, −11u
9
+ 8u
8
+ · · · + 27a − 5, u
10
+
u
7
+ 5u
6
− u
3
+ u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
5
=
0.407407u
9
− 0.296296u
8
+ ··· + 1.18519u + 0.185185
0.148148u
9
+ 0.0740741u
8
+ ··· + 0.703704u − 0.296296
a
6
=
0.407407u
9
− 0.296296u
8
+ ··· + 1.18519u + 0.185185
0.296296u
9
+ 0.148148u
8
+ ··· + 1.40741u − 0.592593
a
9
=
−0.518519u
9
− 0.259259u
8
+ ··· − 0.962963u + 0.0370370
−0.296296u
9
− 0.148148u
8
+ ··· + 0.592593u − 0.407407
a
10
=
−0.814815u
9
− 0.407407u
8
+ ··· − 0.370370u − 0.370370
−0.296296u
9
− 0.148148u
8
+ ··· + 0.592593u − 0.407407
a
2
=
0.407407u
9
− 0.296296u
8
+ ··· + 1.18519u + 0.185185
1
9
u
9
+
5
9
u
8
+ ··· +
7
9
u −
2
9
a
7
=
−u
u
3
+ u
a
1
=
u
0.370370u
9
+ 0.185185u
8
+ ··· + 0.259259u + 0.259259
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
23
9
u
9
−
25
9
u
8
+
10
9
u
7
− 3u
6
−
133
9
u
5
−
116
9
u
4
+
41
9
u
3
+
1
3
u
2
+
19
9
u +
28
9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
10
− 4u
9
+ 6u
8
− 12u
6
+ 15u
5
+ u
4
− 21u
3
+ 25u
2
− 14u + 4
c
2
u
10
+ 6u
9
+ 14u
8
+ 18u
7
+ 22u
6
+ 31u
5
+ 26u
4
+ 7u
3
+ 4u
2
+ 12u + 8
c
3
, c
4
, c
7
u
10
+ u
7
+ 5u
6
− u
3
+ u
2
− u + 1
c
5
, c
8
, c
9
u
10
+ 2u
9
− 7u
8
− 18u
7
+ 9u
6
+ 46u
5
+ 25u
4
− 13u
3
− 10u
2
+ u + 1
c
10
u
10
+ 4u
9
+ ··· − 4u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
10
− 4y
9
+ ··· + 4y + 16
c
2
y
10
− 8y
9
+ ··· − 80y + 64
c
3
, c
4
, c
7
y
10
+ 10y
8
− y
7
+ 27y
6
+ 4y
5
+ 12y
4
+ 9y
3
− y
2
+ y + 1
c
5
, c
8
, c
9
y
10
− 18y
9
+ ··· − 21y + 1
c
10
y
10
+ 8y
9
+ ··· + 1424y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.723110 + 0.623649I
a = 0.401950 + 0.330159I
b = −0.485574 + 1.220240I
−0.90131 − 5.21099I 4.11400 + 8.12783I
u = −0.723110 − 0.623649I
a = 0.401950 − 0.330159I
b = −0.485574 − 1.220240I
−0.90131 + 5.21099I 4.11400 − 8.12783I
u = 0.067084 + 0.694939I
a = 0.43471 + 1.53803I
b = 0.829826 + 0.602084I
−1.96302 + 2.37863I −1.27520 − 1.22709I
u = 0.067084 − 0.694939I
a = 0.43471 − 1.53803I
b = 0.829826 − 0.602084I
−1.96302 − 2.37863I −1.27520 + 1.22709I
u = 0.630715 + 0.297914I
a = 0.574400 − 0.195586I
b = −0.560066 − 0.531210I
1.185420 + 0.648518I 7.38806 − 2.73057I
u = 0.630715 − 0.297914I
a = 0.574400 + 0.195586I
b = −0.560066 + 0.531210I
1.185420 − 0.648518I 7.38806 + 2.73057I
u = −1.034740 + 0.876758I
a = −1.31917 − 0.80288I
b = 1.55315 − 0.33666I
7.82103 − 4.41044I 6.40190 + 3.03613I
u = −1.034740 − 0.876758I
a = −1.31917 + 0.80288I
b = 1.55315 + 0.33666I
7.82103 + 4.41044I 6.40190 − 3.03613I
u = 1.06005 + 1.17909I
a = −1.091890 + 0.674915I
b = 1.66266 + 0.40960I
6.19490 + 11.16340I 4.37125 − 6.32339I
u = 1.06005 − 1.17909I
a = −1.091890 − 0.674915I
b = 1.66266 − 0.40960I
6.19490 − 11.16340I 4.37125 + 6.32339I
5
II. I
u
2
= hu
2
+ b + 1, u
3
+ a + 2u + 1, u
5
+ 2u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
5
=
−u
3
− 2u − 1
−u
2
− 1
a
6
=
−u
3
− 2u − 1
−u
2
− 2
a
9
=
u
4
+ u
2
+ u − 2
u
4
+ 2u
2
+ u
a
10
=
2u
4
+ 3u
2
+ 2u − 2
u
4
+ 2u
2
+ u
a
2
=
u
3
+ 2u + 1
u
4
+ 2u
2
+ 1
a
7
=
−u
u
3
+ u
a
1
=
u
u
4
+ 2u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
4
− 8u
3
− u
2
− 13u − 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ u
4
− u
3
− 2u
2
+ u + 1
c
2
u
5
+ u
4
− 2u
3
− u
2
+ u + 1
c
3
u
5
+ 2u
3
+ u
2
+ 1
c
4
, c
7
u
5
+ 2u
3
− u
2
− 1
c
5
, c
9
u
5
+ u
3
+ 2u
2
+ 1
c
6
u
5
− u
4
− u
3
+ 2u
2
+ u − 1
c
8
u
5
+ u
3
− 2u
2
− 1
c
10
u
5
− 3u
4
+ 7u
3
− 8u
2
+ 5u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
5
− 3y
4
+ 7y
3
− 8y
2
+ 5y −1
c
2
y
5
− 5y
4
+ 8y
3
− 7y
2
+ 3y −1
c
3
, c
4
, c
7
y
5
+ 4y
4
+ 4y
3
− y
2
− 2y −1
c
5
, c
8
, c
9
y
5
+ 2y
4
+ y
3
− 4y
2
− 4y −1
c
10
y
5
+ 5y
4
+ 11y
3
+ 9y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.859460
a = 1.35378
b = −1.73867
3.55538 12.9680
u = 0.300574 + 0.700535I
a = −1.18578 − 1.24715I
b = −0.599596 − 0.421125I
−1.84330 + 3.45949I −2.16713 − 7.95950I
u = 0.300574 − 0.700535I
a = −1.18578 + 1.24715I
b = −0.599596 + 0.421125I
−1.84330 − 3.45949I −2.16713 + 7.95950I
u = 0.12916 + 1.40912I
a = −0.491105 − 0.090789I
b = 0.968932 − 0.363992I
−4.86920 − 1.42206I 0.68335 + 4.57040I
u = 0.12916 − 1.40912I
a = −0.491105 + 0.090789I
b = 0.968932 + 0.363992I
−4.86920 + 1.42206I 0.68335 − 4.57040I
9
III. I
u
3
= h−3646u
11
+ 4692u
10
+ · · · + 3395b + 12871, 24747u
11
−
25539u
10
+ · · · + 16975a − 130862, u
12
− u
11
+ · · · − 6u − 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
5
=
−1.45785u
11
+ 1.50451u
10
+ ··· + 14.1214u + 7.70910
1.07393u
11
− 1.38203u
10
+ ··· − 8.88218u − 3.79116
a
6
=
−1.45785u
11
+ 1.50451u
10
+ ··· + 14.1214u + 7.70910
0.867570u
11
− 1.01290u
10
+ ··· − 7.70427u − 3.83782
a
9
=
3.86074u
11
− 5.29791u
10
+ ··· − 30.9502u − 11.2213
−1.06957u
11
+ 1.43281u
10
+ ··· + 8.75287u + 3.35647
a
10
=
2.79116u
11
− 3.86510u
10
+ ··· − 22.1973u − 7.86480
−1.06957u
11
+ 1.43281u
10
+ ··· + 8.75287u + 3.35647
a
2
=
3.35647u
11
− 4.42604u
10
+ ··· − 25.6789u − 11.3859
−1.07393u
11
+ 1.38203u
10
+ ··· + 8.88218u + 4.79116
a
7
=
−u
u
3
+ u
a
1
=
3.74451u
11
− 5.02480u
10
+ ··· − 29.1594u − 12.4070
−1.56960u
11
+ 1.99193u
10
+ ··· + 13.2389u + 6.02292
(ii) Obstruction class = −1
(iii) Cusp Shapes =
9904
2425
u
11
−
13668
2425
u
10
+
25616
2425
u
9
−
40328
2425
u
8
+
74912
2425
u
7
−
59884
2425
u
6
+
22984
485
u
5
−
3816
97
u
4
+
18152
2425
u
3
−
12692
2425
u
2
−
15928
485
u −
20814
2425
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
3
+ u
2
− 1)
4
c
2
(u
2
− u − 1)
6
c
3
, c
4
, c
7
u
12
− u
11
+ 2u
10
− 3u
9
+ 6u
8
− 3u
7
+ 9u
6
− 5u
5
− 2u
4
− 8u
2
− 6u − 1
c
5
, c
8
, c
9
u
12
+ u
11
+ ··· − 46u − 19
c
10
(u
3
+ u
2
+ 2u + 1)
4
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
3
− y
2
+ 2y −1)
4
c
2
(y
2
− 3y + 1)
6
c
3
, c
4
, c
7
y
12
+ 3y
11
+ ··· − 20y + 1
c
5
, c
8
, c
9
y
12
− 9y
11
+ ··· + 240y + 361
c
10
(y
3
+ 3y
2
+ 2y −1)
4
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.384581 + 0.967717I
a = 0.472201 + 0.655526I
b = 0.618034
−0.92371 + 2.82812I 5.50976 − 2.97945I
u = 0.384581 − 0.967717I
a = 0.472201 − 0.655526I
b = 0.618034
−0.92371 − 2.82812I 5.50976 + 2.97945I
u = 1.17224
a = 1.13192
b = −1.61803
2.83439 −1.01950
u = −0.176090 + 1.382660I
a = −0.566384 + 0.405556I
b = 0.618034
−5.06130 −6 − 1.019511 + 0.10I
u = −0.176090 − 1.382660I
a = −0.566384 − 0.405556I
b = 0.618034
−5.06130 −6 − 1.019511 + 0.10I
u = −0.517507 + 0.159859I
a = −0.95090 + 2.42302I
b = 0.618034
−0.92371 − 2.82812I 5.50976 + 2.97945I
u = −0.517507 − 0.159859I
a = −0.95090 − 2.42302I
b = 0.618034
−0.92371 + 2.82812I 5.50976 − 2.97945I
u = −0.92154 + 1.14616I
a = 1.017000 + 0.670899I
b = −1.61803
6.97197 − 2.82812I 5.50976 + 2.97945I
u = −0.92154 − 1.14616I
a = 1.017000 − 0.670899I
b = −1.61803
6.97197 + 2.82812I 5.50976 − 2.97945I
u = 1.26955 + 0.96884I
a = 1.014420 − 0.568969I
b = −1.61803
6.97197 − 2.82812I 5.50976 + 2.97945I
13
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.26955 − 0.96884I
a = 1.014420 + 0.568969I
b = −1.61803
6.97197 + 2.82812I 5.50976 − 2.97945I
u = −0.250219
a = 3.89540
b = −1.61803
2.83439 −1.01950
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
− 1)
4
(u
5
+ u
4
− u
3
− 2u
2
+ u + 1)
· (u
10
− 4u
9
+ 6u
8
− 12u
6
+ 15u
5
+ u
4
− 21u
3
+ 25u
2
− 14u + 4)
c
2
(u
2
− u − 1)
6
(u
5
+ u
4
− 2u
3
− u
2
+ u + 1)
· (u
10
+ 6u
9
+ 14u
8
+ 18u
7
+ 22u
6
+ 31u
5
+ 26u
4
+ 7u
3
+ 4u
2
+ 12u + 8)
c
3
(u
5
+ 2u
3
+ u
2
+ 1)(u
10
+ u
7
+ 5u
6
− u
3
+ u
2
− u + 1)
· (u
12
− u
11
+ 2u
10
− 3u
9
+ 6u
8
− 3u
7
+ 9u
6
− 5u
5
− 2u
4
− 8u
2
− 6u − 1)
c
4
, c
7
(u
5
+ 2u
3
− u
2
− 1)(u
10
+ u
7
+ 5u
6
− u
3
+ u
2
− u + 1)
· (u
12
− u
11
+ 2u
10
− 3u
9
+ 6u
8
− 3u
7
+ 9u
6
− 5u
5
− 2u
4
− 8u
2
− 6u − 1)
c
5
, c
9
(u
5
+ u
3
+ 2u
2
+ 1)
· (u
10
+ 2u
9
− 7u
8
− 18u
7
+ 9u
6
+ 46u
5
+ 25u
4
− 13u
3
− 10u
2
+ u + 1)
· (u
12
+ u
11
+ ··· − 46u − 19)
c
6
(u
3
+ u
2
− 1)
4
(u
5
− u
4
− u
3
+ 2u
2
+ u − 1)
· (u
10
− 4u
9
+ 6u
8
− 12u
6
+ 15u
5
+ u
4
− 21u
3
+ 25u
2
− 14u + 4)
c
8
(u
5
+ u
3
− 2u
2
− 1)
· (u
10
+ 2u
9
− 7u
8
− 18u
7
+ 9u
6
+ 46u
5
+ 25u
4
− 13u
3
− 10u
2
+ u + 1)
· (u
12
+ u
11
+ ··· − 46u − 19)
c
10
(u
3
+ u
2
+ 2u + 1)
4
(u
5
− 3u
4
+ 7u
3
− 8u
2
+ 5u − 1)
· (u
10
+ 4u
9
+ ··· − 4u + 16)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
3
− y
2
+ 2y −1)
4
(y
5
− 3y
4
+ 7y
3
− 8y
2
+ 5y −1)
· (y
10
− 4y
9
+ ··· + 4y + 16)
c
2
(y
2
− 3y + 1)
6
(y
5
− 5y
4
+ 8y
3
− 7y
2
+ 3y −1)
· (y
10
− 8y
9
+ ··· − 80y + 64)
c
3
, c
4
, c
7
(y
5
+ 4y
4
+ 4y
3
− y
2
− 2y −1)
· (y
10
+ 10y
8
− y
7
+ 27y
6
+ 4y
5
+ 12y
4
+ 9y
3
− y
2
+ y + 1)
· (y
12
+ 3y
11
+ ··· − 20y + 1)
c
5
, c
8
, c
9
(y
5
+ 2y
4
+ y
3
− 4y
2
− 4y −1)(y
10
− 18y
9
+ ··· − 21y + 1)
· (y
12
− 9y
11
+ ··· + 240y + 361)
c
10
(y
3
+ 3y
2
+ 2y −1)
4
(y
5
+ 5y
4
+ 11y
3
+ 9y −1)
· (y
10
+ 8y
9
+ ··· + 1424y + 256)
16
